running laps

At 12:00 noon, Anne, Beth and Carmen begin running laps around a circular
track of length three hundred meters, all starting from the same point on the
track. Each jogger maintains a constant speed in one of the two possible directions
for an indefinite period of time. Show that if Anne’s speed is different
from the other two speeds, then at some later time Anne will be at least one
hundred meters from each of the other runners. (Here, distance is measured
along the shorter of the two arcs separating two runners.)